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How do we prove that among all closed surfaces with a given surface area, the sphere is the one that encloses the largest volume, and not do it by cases?

so far I've tried is that I know the formula for the surface of the sphere and volume of sphere

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You certainly cannot prove this result simply by examining a bunch of closed surfaces and comparing volume enclosed. There are infinitely many different closed surfaces, so you cannot test them all. The problem is not entirely trivial (compare with the 2-dimensional case, with closed simple curves, area, and length). –  Arturo Magidin Jul 19 '11 at 20:02
The is a really good elementary discussion of this problem at cut-the-knot.org/do_you_know/isoperimetric.shtml –  John M Jul 20 '11 at 19:03
Perhaps use the calculus of variations and the Euler-Lagrange formula. –  asmeurer Nov 14 '12 at 6:12
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